Multiphoton non-local quantum interference controlled by an undetected photon

The interference of quanta lies at the heart of quantum physics. The multipartite generalization of single-quanta interference creates entanglement, the coherent superposition of states shared by several quanta. Entanglement allows non-local correlations between many quanta and hence is a key resource for quantum information technology. Entanglement is typically considered to be essential for creating non-local quantum interference. Here, we show that this is not the case and demonstrate multiphoton non-local quantum interference that does not require entanglement of any intrinsic properties of the photons. We harness the superposition of the physical origin of a four-photon product state, which leads to constructive and destructive interference with the photons’ mere existence. With the intrinsic indistinguishability in the generation process of photons, we realize four-photon frustrated quantum interference. This allows us to observe the following noteworthy difference to quantum entanglement: We control the non-local multipartite quantum interference with a photon that we never detect, which does not require quantum entanglement. These non-local properties pave the way for the studies of foundations of quantum physics and potential applications in quantum technologies.

To test the interference of sources I and III, we block the pump P2, fix the position of M2 (φ p ), and perform a coarse scan of the phases of down-converted photons (φ i , φ s1 ) until the interference fringe of two sources emerges. As shown in Supplementary Fig. 2(a), the envelope indicates about 0.2-mm coherent length of the down-converted photons. Then we fix M3 (φ s1 ) at the place where the visibility of interference is maximum and finely tune the phase φ i to obtain the interference pattern. The result is shown in Supplementary Fig. 2(b). The visibility of two-photon coincidence is 95.5%. For sources II and IV, we carry out the same operations. The result is shown in Supplementary  Fig. 2(c) with a visibility of 95.0%. The error bar is smaller than the data point and is not shown here.
The high visibility of two-photon frustrated interference (FI) ensures the path identity, which is essential for observing the interference of four sources. We also observe single-photon FI with high visibilities, showing highlevel indistinguishability of the photons on the same path, as shown in Supplementary Fig. 3

Supplementary Note 3 -Spatial alignment and interference visibility
In this section, we discuss the causes of the reduced visibility of FI. We start from the two-photon FI discussed above. The misalignment of photons on the same path (i1/s1 and i3/s3) and the additional loss for photons i1 and s1 on the optical elements give rise to the different coupling efficiency for sources I and III. Therefore, the collected intensity of source I is lower than that of source III, which is the main reason for the limited visibility. This can be seen in Supplementary Table 1, where the two-fold coincidence counts of sources I and II are lower than that of sources III and IV. If we model the different intensity as transmissivity T, the quantum state of sources I and III is where T 1 is the transmissivity and R 1 is the reflectivity of source I. Photons in the second term are dissipated and not detected. |11 i1s1 and |11 i3s3 are indistinguishable and will interfere. The visibility of Supplementary Eq.(9) is , from which we can estimate T 1 is about 0.737 with V = 95.5%. For sources II and IV, we have a similar analysis and get T 2 = 0.724 with V = 95.0%.
For the four-photon FI, as the beam spacing and parallelism from the BDs are different, the coupling efficiencies of i1 and i2 reduce significantly after the swapping, which aggravates the intensity imbalance. Therefore, we have to make a compromise between the different sources. The result is shown in Supplementary Table 1. q i is the ratio of intensity after and before the swapping of source i. Considering that both T 1 and T 2 above will furthermore reduce the visibility of four-fold coincidence, the quantum state of four-photon FI is where the terms that do not contribute to the four-fold coincidence are omitted, and only the two terms that interfere remain. estimated visibility from independent two-fold coincidence counts is close to the experiment result of 75.47% as in the main text. We estimate the reduction of visibility from higher order emission is about 10% from independent measurement. For comparison, we reduce α to 0.411, as shown in Supplementary Table 2 and measured the interference pattern for four-photon coincidence counts again. We find the four-photon interference visibility decreases to 67.2% as shown in Supplementary Fig. 4. The result is consistent with our theoretical prediction V = 70.32% with α = 0.411. Supplementary Fig. 4. Result of four-fold coincidence counts. The horizontal axis represents the position of M3 (φs1). The fitting curve has visibility of 67.2% and a period of 418.9 nm. The error of visibility calculated from Poisson statistics is 4.50%.

Supplementary Note 4 -Time control of four-photon interference
The temporal indistinguishability for photons on the same path is essential for the four-photon interference 1 . In this section, we analyze the time of the photons generated from different crystals. The lengths of important parts are labeled in Supplementary Fig. 5.
We start our discussion from the two-photon interference scheme, where the QWP is fixed at 0 • . The interference of sources I and II occurs when the pump P3 and the photons s1, i1 arrive at the BBO crystal simultaneously. That's where represent the time experienced by the photons i1, s1, and pump P3, respectively. The beginning point of time is when the pump incident onto BD1, where the pump splits into two paths, as denoted in Supplementary Fig. 5 with "start point". As the position of M2 is fixed (l sp2 + l cp ), Supplementary Eq. (11) could be rewritten as Thus, we should adjust M1 and M3 to meet the above conditions. For sources II and IV, we have similar conditions: When we swap i1 and i2, the time the photons experience are as follows: As long as the conditions of Supplementary Eq. (15) -(18) satisfies, which could be realized by keeping M2, M3, M4 unchanged and scanning the position of M1, the following equations of time indistinguishability still hold: The above analysis shows that, though the photons from the sources I to IV are generated asynchronously 1 , we still can't distinguish the photon on the same path by time, which is essential for the four-photon FI.

Supplementary Note 5 -Two-fold coincidence counts in the four-photon frustrated interference
We also analyze the two-fold coincidence counts in the four-photon interference experiment. The result is shown in Supplementary Fig. 6. There are six two-fold coincidences of the four photons on modes 1 to 4. Only the coincidences on detectors 1 and 3 or detectors 2 and 4 will show the interference pattern, as shown in Supplementary Fig. 6(a). The interference is a result of four-photon FI. Affected by the noise from the first order and second order, as shown in Eq. (1), the interference visibility is limited. The coincidence counts of detectors 2, 3 (source I), detectors 1, 4 (source II), detectors 1, 2 (source III) and detectors 3, 4 (source IV) show no interference as shown in Supplementary  Fig. 6(b). In this section, we give the function used for fitting the V-T correlation in the main text. Supplementary Eq. (10) gives the final state collected by detectors 1-4. When we reduce the transmissivity T of photon s2, it should be rewritten as: where we have denoted identical photons with the same subscript. R is the reflectivity of photon s2. The second term is dissipated and not detected. Therefore, the second term will not contribute to four-fold coincidence but to coincidences on detectors 1, 2, and 3. The visibility of the above equation for four-fold coincidence is We consider α as a variable and are used to quantify the path identity of the photons or the intensity imbalance of the sources, as stated above. Supplementary Fig. 7 shows the four-photon FI with different transmissivities of photon s2, which corresponds to Fig. 3c. As T is reduced, the visibility tends to vanish. The α calculated from coincidence counts (0.521) is higher than that from the fitting curve (0.42) with Supplementary Eq. (28) . The difference may come from the limited long-time stability of our experiment and multiphoton noise from high-order emission.
Supplementary Fig. 7. The relationship between the visibility of four-photon coincidence and the transmissivity of photon s2.
For the coincidences on detectors 1, 2, and 3, as the second term in Supplementary Eq. (27) and term |1210 in Eq. (1) will contribute a constant noise, the visibility is V is proportion to T. Supplementary Fig. 8 shows the four-photon FI with different transmissivities of photon s2, which corresponds to Fig. 4c. As T is reduced, the visibility again tends to vanish.

Supplementary Note 7 -Space-time diagram of non-local and local quantum interference
Throughout our manuscript, we use the terms 'non-local' by its operational definitions. Following the strict Einstein's locality conditions, when we say events X and Y are non-local, it means X lies outside of both the future and past lightcones of Y.
The quantum interference in the frustrated two-photon generation is the interference between photon sources from two crystals. Based on the above definition of non-local, as the twin photons are generated in the same place, the phase setting event a is always in the past light cone of the detection event B (see Supplementary Fig. 9b below). Therefore, we call it local quantum interference.
The analogy between the two-photon entanglement and four-photon frustrated interference is twofold: 1. From the space-time configuration perspective: While the four-photon frustrated interference expands the source into a larger space by swapping the idlers, the phase-setting event a (phase α) can be space-like separated from the detection event B. We call this non-local quantum interference ( Supplementary Fig. 9c below). This situation for the four-photon frustrated interference is similar to the two-photon entanglement ( Supplementary Fig. 9a below), in which the setting events a/b can be space-like separated from the detection events B/A. Supplementary Fig. 8. The relationship between the visibility of three-photon coincidence and the transmissivity of photon s2. Fig. 9a below), when Alice and Bob change their measurement settings, their single counts remain constant, while the coincidence between them shows dependence on both settings. In the four-photon frustrated interference ( Supplementary Fig. 9c below), when Alice and Bob change their phase settings (α and β), their local two-photon coincidence counts (photons 1 and 2, photons 3 and 4, respectively) remain constant, while the four-photon coincidence shows the interference depending on both settings. In particular, when Alice and Bob set the phase α + β = π and keep them unchanged, there will be no four-photon coincidence counts. Therefore, when Alice detects two photons (event A), she can state that there must be no two-photon coincidence counts at Bob's side (event B); that is, event A prohibits event B with the phase setting α + β = π, no matter how far Lab1 and Lab2 are separated from each other. We call this photon-count conditional probability between Alice and Bob the non-local quantum interference. So far, in our experiment, a single laser is the pump source for all of the photon pair generations -and this could open the loophole at the time of the creation of the photon pairs in the lower layer, the phase setting is already fixed. However, this is only done for technical reasons. In principle, the phenomenon we have observed could be demonstrated under strict Einstein locality conditions 2,3 . There are several possible implementations, and we explain two of them now, referring to Fig. 1c: (1) If the pump source is a pulsed laser, then the phases α and β can be set randomly after the pump pulse goes through the two crystals in the lower layer.

From the experimental observation perspective: For an EPR state (Supplementary
(2) The pump sources could be four independent phase-stable lasers. In that way, again it is possible to separate the four sources and the phases α and β, such that the phases are set only after the photon pairs are produced in the lower layer.
Note this requires that the phase of Alice and Bob should be in the same reference frame of the pump light, which may potentially open certain loopholes, such as the freedom-of-choice loophole, especially when one consider the deterministic model. See details in [PNAS 107, 19708 (2010)] 4 . However, to close all loopholes is beyond the scope of the current work and we plan to address this interesting topic in the future. Additionally, investigating the relations of these experimental design to other loopholes will be an interesting theoretical study 5 . To close the locality loophole, we can introduce a random phase on the pump, and configure the phase setting event for Alice/Bob outside the light cone of the phase setting event for the pump.
Supplementary Note 8 -Cancel the noise of three-photon interference Here we give an example of source configuration to improve the three-photon interference visibility. In the main text, it is shown that the maximal three-photon interference visibility is 50% because of the constant noise term |1, 0, 1, 2 produced by a simultaneous event in crystals II and IV.
This, however, is not a fundamental limit, as the additional noise contribution can be canceled by destructive interference (Supplementary Fig. 10). Let's first understand this via the graph representation of our experiment. As shown in Fig. S10a below, each vertex represents the path mode of photons. Each blue edge represents a photon pair source, (i.e., a non-linear crystal), which is labeled from I to IV as shown in Fig. 2. The noise term reducing the three-photon interference (photons 1, 3, and 4 in our experiment) is |1, 0, 1, 2 . So we have to make revisions in the setup in modes 1, 3, and 4. We now add four more crystals: V, VI, VII, and VIII. Crystal V emits non-collinear pairs into modes 1 and 3. Crystal VI produces two photons collinearly in path 4. They together produce an additional three-fold detection in the form |1, 0, 1, 2 . If we set the phase of the light that pumps crystal VI to π with respect to the light that pumps crystal I, then the noise contribution cancels with the newly created term.
At this stage, two additional terms emerge from a combination of crystal V with crystal II and with crystal IV. However, those contributions can also be canceled by adding new crystals VII and VIII that produce two photons collinearly in paths 1 and 3, respectively. Therefore, all noise contributions cancel and the resulting three-photon interference has 100% visibility. Detailed terms (on modes 1, 3, 4) generated from newly added crystals are listed below: Supplementary Note 9 -The Cause of non-local quantum interference As shown in Supplementary Fig. 11, one can generate the GHZ state from a system similar to that of the frustrated four-photon interference, except that we use mode shifters to convert the photons coming from crystals I and II